Seminar on low-dimensional mathematics
24.12.2004
On the hull numbers of torus links
I.Izmestiev
The n-th hull of a union of curves in ${\mathbb R}^3$ is the set of points
$p$ with the following property: Any plane passing through $p$ intersects
the curves at least $2n$ times. The hull number $u(L)$ of a link $L$ is
defined as the minimum number of non-empty hulls a representative of $L$ can
have. Cantarella, G.Kuperberg, Kusner, and Sullivan proved that any
non-trivial knot has hull number at least $2$ and conjectured that the hull
number of the $(p,q)$-torus knot equals its bridge number $\min(p,q)$.
In the talk we will show that the hull numbers of torus links are smaller
than expected, but still large. In particular, for a link of type $(p,p)$ it
is equal to (the ceiling function of) $3p/5$, and in general for the type
$(p,q)$ with $q>p$ it is greater than or equal to $\frac{p}{2}$. Proofs use
the Helly's theorem and certain properties of meridional generators of the
group of a torus link.
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